Multi-type Display Calculus for Semi-De Morgan Logic Fei Liang 1 , 2 - PowerPoint PPT Presentation

Multi-type Display Calculus for Semi-De Morgan Logic Fei Liang 1 , 2 joint work with: G. Greco 1 , M. A. Moshier 3 and A. Palmigiano 1 , 4 1 Delft University of Technology, the Netherlands 2 Sun Yat-sen University, China 3 Chapman University, California, USA 4 University of Johannesburg, South Africa TACL, Prague, 29th June, 2017

Motivation and Aim ◮ Sankappanavar H P . Semi-De Morgan algebras[J]. The Journal of symbolic logic , 1987, 52(3): 712-724 ◮ a common abstraction of De Morgan algebras and distributive pseudo-complemented lattices

Motivation and Aim ◮ Sankappanavar H P . Semi-De Morgan algebras[J]. The Journal of symbolic logic , 1987, 52(3): 712-724 ◮ a common abstraction of De Morgan algebras and distributive pseudo-complemented lattices Boolean • • Heyting a ∧¬ a ≤ 0 • ¬¬ a ≤ a De Morgan • Minimal a ∧ b ≤ c ⇒ a ∧¬ c ≤ ¬ b • a ≤ ¬¬ a quasi-De Morgan • • semi-De Morgan Quasi-Minimal a ≤ ¬¬ a ¬ a = ¬¬¬ a , ¬ 1 = 0 ¬¬ a ∧¬¬ b = ¬¬ ( a ∧ b ) • Preminimal ¬ ( a ∨ b ) = ¬ a ∧¬ b , ¬ 0 = 1

Motivation and Aim ◮ Sankappanavar H P . Semi-De Morgan algebras[J]. The Journal of symbolic logic , 1987, 52(3): 712-724 ◮ a common abstraction of De Morgan algebras and distributive pseudo-complemented lattices Boolean • • Heyting a ∧¬ a ≤ 0 • ¬¬ a ≤ a De Morgan • Minimal a ∧ b ≤ c ⇒ a ∧¬ c ≤ ¬ b • a ≤ ¬¬ a quasi-De Morgan • • semi-De Morgan Quasi-Minimal a ≤ ¬¬ a ¬ a = ¬¬¬ a , ¬ 1 = 0 ¬¬ a ∧¬¬ b = ¬¬ ( a ∧ b ) • Preminimal ¬ ( a ∨ b ) = ¬ a ∧¬ b , ¬ 0 = 1 ◮ Ma M, Liang F. Sequent Calculi for Semi-De Morgan and De Morgan Algebras[J]. arXiv preprint:1611.05231 , 2016.

Motivation and Aim Is there an uniform way to deal with semi De Morgan negation and preserve real subformula property?

Motivation and Aim Is there an uniform way to deal with semi De Morgan negation and preserve real subformula property? ◮ The answer is “Yes”, via multi-type methodology !

Preliminaries

De Morgan and semi-De Morgan Algebras Definition If ( A , ∨ , ∧ , ⊤ , ⊥ ) is a bounded distributive lattice, then an algebra A = ( A , ∨ , ∧ , ¬ , ⊤ , ⊥ ) is: for all a , b ∈ A , De Morgan algebra semi-De Morgan algebra ¬ ( a ∨ b ) = ¬ a ∧¬ b ¬ ( a ∨ b ) = ¬ a ∧¬ b ¬ ( a ∧ b ) = ¬ a ∨¬ b ¬¬ ( a ∧ b ) = ¬¬ a ∧¬¬ b ¬¬ a = a ¬¬¬ a = ¬ a ¬⊥ = ⊤ , ¬⊤ = ⊥ ¬⊥ = ⊤ , ¬⊤ = ⊥

De Morgan and semi-De Morgan Algebras Definition If ( A , ∨ , ∧ , ⊤ , ⊥ ) is a bounded distributive lattice, then an algebra A = ( A , ∨ , ∧ , ¬ , ⊤ , ⊥ ) is: for all a , b ∈ A , De Morgan algebra semi-De Morgan algebra ¬ ( a ∨ b ) = ¬ a ∧¬ b ¬ ( a ∨ b ) = ¬ a ∧¬ b ¬ ( a ∧ b ) = ¬ a ∨¬ b ¬¬ ( a ∧ b ) = ¬¬ a ∧¬¬ b ¬¬ a = a ¬¬¬ a = ¬ a ¬⊥ = ⊤ , ¬⊤ = ⊥ ¬⊥ = ⊤ , ¬⊤ = ⊥ Fact A semi-De Morgan algebra A is a De Morgan algebra if and only if A satisfies the equation a ∨ b = ¬ ( ¬ a ∧¬ b ) = ¬¬ ( a ∨ b ) .

De Morgan and semi-De Morgan Algebras Definition If ( A , ∨ , ∧ , ⊤ , ⊥ ) is a bounded distributive lattice, then an algebra A = ( A , ∨ , ∧ , ¬ , ⊤ , ⊥ ) is: for all a , b ∈ A , De Morgan algebra semi-De Morgan algebra ¬ ( a ∨ b ) = ¬ a ∧¬ b ¬ ( a ∨ b ) = ¬ a ∧¬ b ¬ ( a ∧ b ) = ¬ a ∨¬ b ¬¬ ( a ∧ b ) = ¬¬ a ∧¬¬ b ¬¬ a = a ¬¬¬ a = ¬ a ¬⊥ = ⊤ , ¬⊤ = ⊥ ¬⊥ = ⊤ , ¬⊤ = ⊥ Fact A semi-De Morgan algebra A is a De Morgan algebra if and only if A satisfies the equation a ∨ b = ¬ ( ¬ a ∧¬ b ) = ¬¬ ( a ∨ b ) . ¬¬ ( a ∧ b ) = ¬¬ a ∧¬¬ b and ¬¬¬ a = ¬ a can not be transformed into structural rules immediately!

Stratergy ◮ from semi-De Morgan algebras to construct heterogeneous semi-De Morgan algebras in which every axiom is analytic

Stratergy ◮ from semi-De Morgan algebras to construct heterogeneous semi-De Morgan algebras in which every axiom is analytic ◮ from heterogeneous semi-De Morgan algebras to construct semi-De Morgan algebras

From single type to multi-type

Multi-type enviroment Lemma Given an SM-algebra L = ( L , ∧ , ∨ , ⊤ , ⊥ , ¬ ) , let K := {¬¬ a ∈ L | a ∈ L } . Define h : L ։ K and e : K ֒ → L by the assignments a �→ ¬¬ a and α �→ α , respectively. Then for all α ∈ K and a ∈ L, h ( e ( α )) = α

Multi-type enviroment Definition For any SM-algebra L = ( L , ∧ , ∨ , ⊤ , ⊥ , ¬ ) , let the kernel of L be the algebra K L = ( K , ∩ , ∪ , ∼ , 1 , 0 ) defined as follows: K1. K := Range ( h ) , where h : L ։ K is defined by letting h ( a ) = ¬¬ a for any a ∈ L ; K2. α ∪ β := h ( ¬¬ ( e ( α ) ∨ e ( β ))) for all α,β ∈ K ; K3. α ∩ β := h ( e ( α ) ∧ e ( β )) for all α,β ∈ K ; K4. 1 := h ( ⊤ ) ; K5. 0 := h ( ⊥ ) ; K6. ∼ α := h ( ¬ e ( α )) .

Multi-type enviroment Lemma For any SM-algebra L , 1. the kernel K L is a DM-algebra. 2. h is a lattice-homomorphism from L onto K , and for all α,β ∈ K, e ( α ) ∧ e ( β ) = e ( α ∩ β ) e ( 1 ) = ⊤ e ( 0 ) = ⊥ .

Heterogenous algebra Definition A heterogeneous SDM-algebra (HSM-algebra) is a tuple ( L , A , e , h ) satisfying the following conditions: H1 L is a bounded distributive lattice;

Heterogenous algebra Definition A heterogeneous SDM-algebra (HSM-algebra) is a tuple ( L , A , e , h ) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice;

Heterogenous algebra Definition A heterogeneous SDM-algebra (HSM-algebra) is a tuple ( L , A , e , h ) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice; H3 e : A ֒ → L is an order embedding, which satisfies: for all α 1 ,α 2 ∈ A , e ( α 1 ) ∧ e ( α 2 ) = e ( α 1 ∩ α 2 ) e ( 1 ) = ⊤ e ( 0 ) = ⊥ and and

Heterogenous algebra Definition A heterogeneous SDM-algebra (HSM-algebra) is a tuple ( L , A , e , h ) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice; H3 e : A ֒ → L is an order embedding, which satisfies: for all α 1 ,α 2 ∈ A , e ( α 1 ) ∧ e ( α 2 ) = e ( α 1 ∩ α 2 ) e ( 1 ) = ⊤ e ( 0 ) = ⊥ and and H4 h : L ։ A is a lattice homomorphism;

Heterogenous algebra Definition A heterogeneous SDM-algebra (HSM-algebra) is a tuple ( L , A , e , h ) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice; H3 e : A ֒ → L is an order embedding, which satisfies: for all α 1 ,α 2 ∈ A , e ( α 1 ) ∧ e ( α 2 ) = e ( α 1 ∩ α 2 ) e ( 1 ) = ⊤ e ( 0 ) = ⊥ and and H4 h : L ։ A is a lattice homomorphism; H5 h ( e ( α )) = α for every α ∈ A . e L A h ∼

From multi-type to single type

Heterogenous algebra Lemma If ( L , D , e , h ) is an heterogeneous SM-algebra, then L can be endowed with a structure of SM-algebra defining ¬ : L → L by ¬ a := e ( ∼ h ( a )) for every a ∈ L . Moreover, D � K .

Heterogenous algebra Lemma If ( L , D , e , h ) is an heterogeneous SM-algebra, then L can be endowed with a structure of SM-algebra defining ¬ : L → L by ¬ a := e ( ∼ h ( a )) for every a ∈ L . Moreover, D � K . Definition For any SM-algebra A , we let A + = ( L , K , h , e ) , where: · L is the lattice reduct of A ; · K is the kernel of A ; · e : K ֒ → L is defined by e ( α ) = α for all α ∈ K ; · h : L ։ K is defined by h ( a ) = ¬¬ a for all a ∈ L ; For any HSM-algebra H , we let H + = ( L , ¬ ) where: · L is the distributive lattice of H ; · ¬ : L → L is defined by the assignment a �→ e ( ∼ h ( a )) for all a ∈ L .

Heterogenous representation theory For any SM-algebra A and any HSM-algebra H : A � ( A + ) + H � ( H + ) + . and

Algebraic semantics for multi-type display calculus

Canonical extension Definition A HSM-algebra is perfect if: 1. both L and A are perfect; 2. e is an order-embedding and is completely meet-preserving; 3. h is a complete homomorphism. Corollary If ( L , D , e , h ) is an HSM-algebra, then ( L δ , D δ , e π , h δ ) is a perfect HSM-algebra.

Canonical extension e ′ ⊢ e π L δ A δ h δ ∼ δ ⊣⊢ h ′ e L A h ∼ Corollary If ( L , ¬ ) is an SM-algebra, then L δ can be endowed with the structure of SM-algebra by defining ¬ δ : L δ → L δ by ¬ δ := e π ◦∼ δ ◦ h δ . Moreover, K δ L � K L δ .

Multi-type proper display calculus

Hilbert style semi-De Morgan logic ◮ the language L A ::= p | ⊥ | ⊤ | ¬ A | A ∧ A | A ∨ A ◮ Axioms (A1) ⊥ ⊢ A (A2) A ⊢ ⊤ (A3) ¬⊤ ⊢ ⊥ (A4) ⊤ ⊢ ¬⊥ (A5) A ⊢ A (A6) A ∧ B ⊢ A (A7) A ∧ B ⊢ B (A8) A ⊢ A ∨ B (A9) B ⊢ A ∨ B (A10) ¬ A ⊢ ¬¬¬ A (A11) ¬¬¬ A ⊢ ¬ A (A12) ¬ A ∧¬ B ⊢ ¬ ( A ∨ B ) (A13) ¬¬ A ∧¬¬ B ⊢ ¬¬ ( A ∧ B ) (A14) A ∧ ( B ∨ C ) ⊢ ( A ∧ B ) ∨ ( A ∧ C ) ◮ Rules R1. If A ⊢ B and B ⊢ C , then A ⊢ C ; R2. If A ⊢ B and A ⊢ C , then A ⊢ B ∧ C ; R3. If A ⊢ B and C ⊢ B , then A ∨ C ⊢ B ; R4. If A ⊢ B , then ¬ B ⊢ ¬ A .

Multi-type Display calculus ◮ Structural and operational language of D.DL:  A ::= p | ⊤ | ⊥ | � α | A ∧ A | A ∨ A    DL    X ::= ˆ ⊤ | ˇ � Γ | X ˆ ∧ X | X ˇ ⊥ | ˇ ∨ X | X ˆ > X | X ˇ  → X 